Geo5.3.48

From Example Problems
Jump to: navigation, search
Show that the midpoints of normal chords of {\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}=1\, is [{\frac  {a^{6}}{x^{2}}}-{\frac  {b^{6}}{y^{2}}}][{\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}]^{2}=(a^{2}+b^{2})^{2}\,

Let P be the midpooint of a chord of the given hyperbola,then its equation is

{\frac  {xx_{1}}{a^{2}}}-{\frac  {yy_{1}}{b^{2}}}={\frac  {x_{1}^{{2}}}{a^{2}}}-{\frac  {y_{1}^{{2}}}{b^{2}}}\,

This is a normal to the hyperbola ,hence according to the normal condition

{\frac  {a^{2}}{[{\frac  {x_{1}}{a^{2}}}]^{2}}}-{\frac  {b^{2}}{[{\frac  {y_{1}}{b^{2}}}]^{2}}}={\frac  {(a^{2}+b^{2})^{2}}{[{\frac  {x_{1}^{{2}}}{a^{2}}}-{\frac  {y_{1}^{{2}}}{b^{2}}}]^{2}}}\,

Therefore the locus of P is [{\frac  {a^{6}}{x^{2}}}-{\frac  {b^{6}}{y^{2}}}][{\frac  {x^{2}}{a^{2}}}-{\frac  {y^{2}}{b^{2}}}]^{2}=(a^{2}+b^{2})^{2}\,


Main Page:Geometry:Hyperbola